Properties

Label 1155.2.i.b
Level 1155
Weight 2
Character orbit 1155.i
Analytic conductor 9.223
Analytic rank 0
Dimension 8
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1155.i (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( \beta_{4} - \beta_{5} ) q^{2} \) \( -\beta_{3} q^{3} \) \( + ( -\beta_{2} + \beta_{3} - \beta_{6} ) q^{4} \) \( + \beta_{3} q^{5} \) \( + ( \beta_{1} + \beta_{5} + \beta_{7} ) q^{6} \) \( + ( \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{7} \) \( + ( -\beta_{1} + \beta_{4} ) q^{8} \) \(- q^{9}\) \(+O(q^{10})\) \( q\) \( + ( \beta_{4} - \beta_{5} ) q^{2} \) \( -\beta_{3} q^{3} \) \( + ( -\beta_{2} + \beta_{3} - \beta_{6} ) q^{4} \) \( + \beta_{3} q^{5} \) \( + ( \beta_{1} + \beta_{5} + \beta_{7} ) q^{6} \) \( + ( \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{7} \) \( + ( -\beta_{1} + \beta_{4} ) q^{8} \) \(- q^{9}\) \( + ( -\beta_{1} - \beta_{5} - \beta_{7} ) q^{10} \) \( + ( -3 - \beta_{1} + \beta_{5} ) q^{11} \) \( + ( -1 + \beta_{2} - \beta_{6} ) q^{12} \) \( + ( -3 \beta_{1} - 3 \beta_{5} ) q^{13} \) \( + ( -2 - 2 \beta_{2} + 2 \beta_{3} - \beta_{6} ) q^{14} \) \(+ q^{15}\) \( + ( -1 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} ) q^{16} \) \( + ( \beta_{1} + \beta_{5} + 2 \beta_{7} ) q^{17} \) \( + ( -\beta_{4} + \beta_{5} ) q^{18} \) \( + ( 2 \beta_{1} + 2 \beta_{5} + 2 \beta_{7} ) q^{19} \) \( + ( 1 - \beta_{2} + \beta_{6} ) q^{20} \) \( + ( 3 \beta_{1} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{21} \) \( + ( 1 + \beta_{2} - \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{22} \) \( -4 q^{23} \) \( + \beta_{7} q^{24} \) \(- q^{25}\) \( + ( 3 - 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{6} ) q^{26} \) \( + \beta_{3} q^{27} \) \( + ( -\beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{28} \) \( + ( 2 \beta_{4} - 2 \beta_{5} ) q^{29} \) \( + ( \beta_{4} - \beta_{5} ) q^{30} \) \( + ( 4 - 4 \beta_{2} - 4 \beta_{3} + 4 \beta_{6} ) q^{31} \) \( + ( -4 \beta_{1} - \beta_{4} + 5 \beta_{5} ) q^{32} \) \( + ( -\beta_{1} + 3 \beta_{3} - \beta_{5} ) q^{33} \) \( + ( -1 + \beta_{2} + 3 \beta_{3} - \beta_{6} ) q^{34} \) \( + ( -3 \beta_{1} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{35} \) \( + ( \beta_{2} - \beta_{3} + \beta_{6} ) q^{36} \) \( + ( -4 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} ) q^{37} \) \( + ( -2 + 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{6} ) q^{38} \) \( + ( 3 \beta_{1} - 3 \beta_{5} ) q^{39} \) \( -\beta_{7} q^{40} \) \( + ( -2 \beta_{1} - 2 \beta_{5} + 2 \beta_{7} ) q^{41} \) \( + ( -1 + \beta_{2} + 3 \beta_{3} - 2 \beta_{6} ) q^{42} \) \( + ( -\beta_{1} + 2 \beta_{4} - \beta_{5} ) q^{43} \) \( + ( -\beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{44} \) \( -\beta_{3} q^{45} \) \( + ( -4 \beta_{4} + 4 \beta_{5} ) q^{46} \) \( + ( -4 + 4 \beta_{2} + 4 \beta_{3} - 4 \beta_{6} ) q^{47} \) \( + ( -2 + 2 \beta_{2} + \beta_{3} - 2 \beta_{6} ) q^{48} \) \( + ( -4 \beta_{2} + 3 \beta_{3} - 4 \beta_{6} ) q^{49} \) \( + ( -\beta_{4} + \beta_{5} ) q^{50} \) \( + ( \beta_{1} - 2 \beta_{4} + \beta_{5} ) q^{51} \) \( + ( 3 \beta_{1} + 3 \beta_{5} + 6 \beta_{7} ) q^{52} \) \( + ( 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{6} ) q^{53} \) \( + ( -\beta_{1} - \beta_{5} - \beta_{7} ) q^{54} \) \( + ( \beta_{1} - 3 \beta_{3} + \beta_{5} ) q^{55} \) \( + ( 1 - 2 \beta_{2} + 3 \beta_{3} + \beta_{6} ) q^{56} \) \( + ( -2 \beta_{4} + 2 \beta_{5} ) q^{57} \) \( + ( -4 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} ) q^{58} \) \( + ( -4 + 4 \beta_{2} - 6 \beta_{3} - 4 \beta_{6} ) q^{59} \) \( + ( -\beta_{2} + \beta_{3} - \beta_{6} ) q^{60} \) \( + ( -2 \beta_{1} - 2 \beta_{5} - 4 \beta_{7} ) q^{61} \) \( + ( 12 \beta_{1} + 12 \beta_{5} + 8 \beta_{7} ) q^{62} \) \( + ( -\beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{63} \) \( + ( 4 + \beta_{2} - \beta_{3} + \beta_{6} ) q^{64} \) \( + ( -3 \beta_{1} + 3 \beta_{5} ) q^{65} \) \( + ( 1 - 3 \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{66} \) \( + ( -2 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{6} ) q^{67} \) \( + ( -3 \beta_{1} - 3 \beta_{5} ) q^{68} \) \( + 4 \beta_{3} q^{69} \) \( + ( 1 - \beta_{2} - 3 \beta_{3} + 2 \beta_{6} ) q^{70} \) \( + ( -4 \beta_{2} + 4 \beta_{3} - 4 \beta_{6} ) q^{71} \) \( + ( \beta_{1} - \beta_{4} ) q^{72} \) \( + ( -5 \beta_{1} - 5 \beta_{5} - 4 \beta_{7} ) q^{73} \) \( + ( -2 \beta_{1} - 6 \beta_{4} + 8 \beta_{5} ) q^{74} \) \( + \beta_{3} q^{75} \) \( + ( -4 \beta_{1} - 4 \beta_{5} - 2 \beta_{7} ) q^{76} \) \( + ( 3 + \beta_{3} - 3 \beta_{4} + 6 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{77} \) \( + ( -3 - 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{6} ) q^{78} \) \( + ( 2 \beta_{1} - 2 \beta_{4} ) q^{79} \) \( + ( 2 - 2 \beta_{2} - \beta_{3} + 2 \beta_{6} ) q^{80} \) \(+ q^{81}\) \( + ( 2 - 2 \beta_{2} + 2 \beta_{6} ) q^{82} \) \( + ( -\beta_{1} - \beta_{5} ) q^{83} \) \( + ( -2 \beta_{4} - 3 \beta_{5} - 2 \beta_{7} ) q^{84} \) \( + ( -\beta_{1} + 2 \beta_{4} - \beta_{5} ) q^{85} \) \( + ( -3 - \beta_{2} + \beta_{3} - \beta_{6} ) q^{86} \) \( + ( 2 \beta_{1} + 2 \beta_{5} + 2 \beta_{7} ) q^{87} \) \( + ( -1 + 3 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{6} ) q^{88} \) \( + ( -4 + 4 \beta_{2} + 2 \beta_{3} - 4 \beta_{6} ) q^{89} \) \( + ( \beta_{1} + \beta_{5} + \beta_{7} ) q^{90} \) \( + ( 9 - 6 \beta_{2} - 3 \beta_{3} ) q^{91} \) \( + ( 4 \beta_{2} - 4 \beta_{3} + 4 \beta_{6} ) q^{92} \) \( + ( -4 - 4 \beta_{2} + 4 \beta_{3} - 4 \beta_{6} ) q^{93} \) \( + ( -12 \beta_{1} - 12 \beta_{5} - 8 \beta_{7} ) q^{94} \) \( + ( 2 \beta_{4} - 2 \beta_{5} ) q^{95} \) \( + ( -5 \beta_{1} - 5 \beta_{5} - \beta_{7} ) q^{96} \) \( + ( 4 - 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{6} ) q^{97} \) \( + ( -3 \beta_{1} - 4 \beta_{4} + 9 \beta_{5} + \beta_{7} ) q^{98} \) \( + ( 3 + \beta_{1} - \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 8q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 8q^{9} \) \(\mathstrut -\mathstrut 24q^{11} \) \(\mathstrut -\mathstrut 20q^{14} \) \(\mathstrut +\mathstrut 8q^{15} \) \(\mathstrut -\mathstrut 8q^{16} \) \(\mathstrut +\mathstrut 8q^{22} \) \(\mathstrut -\mathstrut 32q^{23} \) \(\mathstrut -\mathstrut 8q^{25} \) \(\mathstrut -\mathstrut 32q^{37} \) \(\mathstrut +\mathstrut 4q^{42} \) \(\mathstrut -\mathstrut 4q^{56} \) \(\mathstrut -\mathstrut 32q^{58} \) \(\mathstrut +\mathstrut 32q^{64} \) \(\mathstrut -\mathstrut 16q^{67} \) \(\mathstrut -\mathstrut 4q^{70} \) \(\mathstrut +\mathstrut 16q^{77} \) \(\mathstrut -\mathstrut 24q^{78} \) \(\mathstrut +\mathstrut 8q^{81} \) \(\mathstrut -\mathstrut 24q^{86} \) \(\mathstrut -\mathstrut 8q^{88} \) \(\mathstrut +\mathstrut 48q^{91} \) \(\mathstrut -\mathstrut 32q^{93} \) \(\mathstrut +\mathstrut 24q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring:

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \zeta_{24}^{3} \)
\(\beta_{2}\)\(=\)\( \zeta_{24}^{4} + \zeta_{24}^{2} \)
\(\beta_{3}\)\(=\)\( \zeta_{24}^{6} \)
\(\beta_{4}\)\(=\)\( \zeta_{24}^{7} + \zeta_{24} \)
\(\beta_{5}\)\(=\)\( -\zeta_{24}^{5} + \zeta_{24} \)
\(\beta_{6}\)\(=\)\( -\zeta_{24}^{4} + \zeta_{24}^{2} \)
\(\beta_{7}\)\(=\)\( -\zeta_{24}^{7} + \zeta_{24}^{5} \)
\(1\)\(=\)\(\beta_0\)
\(\zeta_{24}\)\(=\)\((\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\)\()/2\)
\(\zeta_{24}^{2}\)\(=\)\((\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{2}\)\()/2\)
\(\zeta_{24}^{3}\)\(=\)\(\beta_{1}\)
\(\zeta_{24}^{4}\)\(=\)\((\)\(-\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{2}\)\()/2\)
\(\zeta_{24}^{5}\)\(=\)\((\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\)\()/2\)
\(\zeta_{24}^{6}\)\(=\)\(\beta_{3}\)
\(\zeta_{24}^{7}\)\(=\)\((\)\(-\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\)\()/2\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1155\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(386\) \(661\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
−0.965926 0.258819i
0.965926 0.258819i
−0.258819 0.965926i
0.258819 0.965926i
0.258819 + 0.965926i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 + 0.258819i
1.93185i 1.00000i −1.73205 1.00000i −1.93185 0.189469 2.63896i 0.517638i −1.00000 1.93185
76.2 1.93185i 1.00000i −1.73205 1.00000i 1.93185 −0.189469 2.63896i 0.517638i −1.00000 −1.93185
76.3 0.517638i 1.00000i 1.73205 1.00000i −0.517638 −2.63896 + 0.189469i 1.93185i −1.00000 0.517638
76.4 0.517638i 1.00000i 1.73205 1.00000i 0.517638 2.63896 + 0.189469i 1.93185i −1.00000 −0.517638
76.5 0.517638i 1.00000i 1.73205 1.00000i 0.517638 2.63896 0.189469i 1.93185i −1.00000 −0.517638
76.6 0.517638i 1.00000i 1.73205 1.00000i −0.517638 −2.63896 0.189469i 1.93185i −1.00000 0.517638
76.7 1.93185i 1.00000i −1.73205 1.00000i 1.93185 −0.189469 + 2.63896i 0.517638i −1.00000 −1.93185
76.8 1.93185i 1.00000i −1.73205 1.00000i −1.93185 0.189469 + 2.63896i 0.517638i −1.00000 1.93185
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.8
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.b Odd 1 no
11.b Odd 1 no
77.b Even 1 no

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{4} \) \(\mathstrut +\mathstrut 4 T_{2}^{2} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{2}^{\mathrm{new}}(1155, [\chi])\).